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Chapter 2: Problem 30

Simplify the given expression. $$ \frac{\left(x^{4} y^{3 / 4}\right)^{-3}}{\left(x^{5} y^{-2}\right)^{4}} $$

Short Answer

Expert verified
The simplified expression is \(x^{-32} y^{\frac{23}{4}}\).

Step by step solution

01

Apply the Power Rule to each term inside the parentheses

Using the Power Rule, we raise each term inside the parentheses to the given power, which means that we multiply the exponents. Let's apply this rule to the numerator and the denominator of the expression: $$ \frac{\left(x^{4} y^{\frac{3}{4}}\right)^{-3}}{\left(x^{5} y^{-2}\right)^{4}} = \frac{x^{4 \cdot (-3)} y^{\frac{3}{4} \cdot (-3)}}{x^{5 \cdot 4} y^{-2 \cdot 4}} $$
02

Simplify the exponents

Now, we can simplify the exponents by performing the multiplication: $$ \frac{x^{-12} y^{-\frac{9}{4}}}{x^{20} y^{-8}} $$
03

Use the Quotient Rule

To simplify the expression further, we can use the Quotient Rule in exponents, which states that \(a^{m} \div a^{n} = a^{m-n}\). This allows us to subtract the exponents of like terms in the numerator and the denominator: $$ \frac{x^{-12} y^{-\frac{9}{4}}}{x^{20} y^{-8}} = x^{-12-20} y^{-\frac{9}{4}-(-8)} $$
04

Simplify the final expression

Now, we can perform the subtraction to obtain the simplified expression: $$ x^{-32} y^{\frac{23}{4}} $$ Our final simplified expression is: $$ \boxed{x^{-32} y^{\frac{23}{4}}} $$

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Most popular questions from this chapter

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